#392228

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,19,13,26)(11,20,22,16,23,31,27,30)(12,17,24,33,25,34,28,32)(21,36,39,40,38,35,37,29);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,31)(19,30)(20,26)(24,39)(25,38)(28,37)(32,35)(33,36)(34,40);(9,12)(10,17)(11,21)(13,28)(14,24)(15,25)(16,29)(18,33)(19,34)(20,35)(22,39)(23,38)(26,32)(27,37)(30,40)(31,36)

Polycyclic presentation:

2^9|a^2=h,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299424);